Steps With an End In Sight

Use properties of equality to isolate variables and solve algebraic equations

There are some equations that you can solve in your head quickly, but other equations are more complicated. Multi-step equations, ones that takes several steps to solve, can still be simplified and solved by applying basic algebraic rules such as the multiplication and addition properties of equality.

In this section we will explore methods for solving multi-step equations that contain grouping symbols and several mathematical operations. We will also learn techniques for solving multi-step equations that contain absolute values. Finally, we will learn that some equations have no solutions, while others have an infinite number of solutions.

First, let’s define some important terminology:

• variables:  variables are symbols that stand for an unknown quantity, they are often represented with letters, like x, y, or z.
• coefficient: Sometimes a variable is multiplied by a number. This number is called the coefficient of the variable. For example, the coefficient of 3x is 3.
• term: a single number, or variables and numbers connected by multiplication. -4, 6x and [latex]x^2[/latex] are all terms
• expression: groups of terms connected by addition and subtraction.  [latex]2x^2-5[/latex] is an expression
• equation:  an equation is a mathematical statement that two expressions are equal. An equation will always contain an equal sign with an expression on each side. Think of an equal sign as meaning “the same as.” Some examples of equations are [latex]y = mx +b\\[/latex],  [latex]\frac{3}{4}r = v^{3} - r\\[/latex], and  [latex]2(6-d) + f(3 +k) = \frac{1}{4}d\\[/latex]

The following figure shows how coefficients, variables, terms, and expressions all come together to make equations. In the equation [latex]2x-3^2=10x[/latex], the variable is [latex]x[/latex], a coefficient is [latex]10[/latex], a term is [latex]10x[/latex], an expression is [latex]2x-3^2[/latex].

There are some equations that you can solve in your head quickly. For example—what is the value of y in the equation [latex]2y=6[/latex]? Chances are you didn’t need to get out a pencil and paper to calculate that [latex]y=3[/latex]. You only needed to do one thing to get the answer: divide 6 by 2.

Other equations are more complicated. Solving [latex]\displaystyle 4\left( \frac{1}{3}t+\frac{1}{2}\right)=6[/latex] without writing anything down is difficult! That’s because this equation contains not just a variable but also fractions and terms inside parentheses. This is a multi-step equation, one that takes several steps to solve. Although multi-step equations take more time and more operations, they can still be simplified and solved by applying basic algebraic rules.

Remember that you can think of an equation as a balance scale, with the goal being to rewrite the equation so that it is easier to solve but still balanced. The addition property of equality and the multiplication property of equality explain how you can keep the scale, or the equation, balanced. Whenever you perform an operation to one side of the equation, if you perform the same exact operation to the other side, you’ll keep both sides of the equation equal.

If the equation is in the form [latex]ax+b=c[/latex], where x is the variable, you can solve the equation as before. First “undo” the addition and subtraction, and then “undo” the multiplication and division.

Some equations may have the variable on both sides of the equal sign, as in this equation: [latex]4x-6=2x+10[/latex].

To solve this equation, we need to “move” one of the variable terms. This can make it difficult to decide which side to work with. It doesn’t matter which term gets moved, [latex]4x[/latex] or [latex]2x[/latex], however, to avoid negative coefficients, you can move the smaller term.

In this video, we show an example of solving equations that have variables on both sides of hte equal sign.

Solving Multi-Step Equations With Absolute Value

We can apply the same techniques we used for solving a one-step equation which contained absolute value to an equation that will take more than one step to solve.  Let’s start with an example where the first step is to write two equations, one equal to positive 26 and one equal to negative 26.

In the next video, we show more examples of solving a simple absolute value equation.

Now let’s look at an example where you need to do an algebraic step or two before you can write your two equations. The goal here is to get the absolute value on one side of the equation by itself. Then we can proceed as we did in the previous example.

In the two videos that follow, we show examples of how to solve an absolute value equation that requires you to isolate the absolute value first using mathematical operations.

The Distributive Property

As we solve linear equations, we often need to do some work to write the linear equations in a form we are familiar with solving. This section will focus on manipulating an equation we are asked to solve in such a way that we can use the skills we learned for solving multi-step equations to ultimately arrive at the solution.

Parentheses can make solving a problem difficult, if not impossible. To get rid of these unwanted parentheses we have the distributive property. Using this property we multiply the number in front of the parentheses by each term inside of the parentheses.

In the video that follows, we show another example of how to use the distributive property to solve a multi-step linear equation.

In the next example, you will see that there are parentheses on both sides of the equal sign, so you will need to use the distributive property twice. Notice that you are going to need to distribute a negative number, so be careful with negative signs!

In the following video, we solve another multi-step equation with two sets of parentheses.

Sometimes, you will encounter a multi-step equation with fractions. If you prefer not working with fractions, you can use the multiplication property of equality to multiply both sides of the equation by a common denominator of all of the fractions in the equation. This will clear all the fractions out of the equation. See the example below.

Of course, if you like to work with fractions, you can just apply your knowledge of operations with fractions and solve.

In the following video, we show how to solve a multi-step equation with fractions.

Regardless of which method you use to solve equations containing variables, you will get the same answer. You can choose the method you find the easiest! Remember to check your answer by substituting your solution into the original equation.

Sometimes, you will encounter a multi-step equation with decimals. If you prefer not working with decimals, you can use the multiplication property of equality to multiply both sides of the equation by a a factor of 10 that will help clear the decimals. See the example below.

In the following video, we show another example of clearing decimals first to solve a multi-step linear equation.

Here are some steps to follow when you solve multi-step equations.

Classify Solutions to Linear Equations

There are three cases that can come up as we are solving linear equations. We have already seen one, where an equation has one solution. Sometimes we come across equations that don’t have any solutions, and even some that have an infinite number of solutions. The case where an equation has no solution is illustrated in the next examples.

Equations with no solutions

This is not a solution! You did not find a value for x. Solving for x the way you know how, you arrive at the false statement [latex]4=5[/latex]. Surely 4 cannot be equal to 5!

This may make sense when you consider the second line in the solution where like terms were combined. If you multiply a number by 2 and add 4 you would never get the same answer as when you multiply that same number by 2 and add 5. Since there is no value of x that will ever make this a true statement, the solution to the equation above is “no solution.”

Be careful that you do not confuse the solution [latex]x=0[/latex] with “no solution.” The solution [latex]x=0[/latex] means that the value 0 satisfies the equation, so there is a solution. “No solution” means that there is no value, not even 0, which would satisfy the equation.

Also, be careful not to make the mistake of thinking that the equation [latex]4=5[/latex] means that 4 and 5 are values for x that are solutions. If you substitute these values into the original equation, you’ll see that they do not satisfy the equation. This is because there is truly no solution—there are no values for x that will make the equation [latex]12+2x–8=7x+5–5x[/latex] true.

Algebraic Equations with an Infinite Number of Solutions

You have seen that if an equation has no solution, you end up with a false statement instead of a value for x. It is possible to have an equation where any value for x will provide a solution to the equation. In the example below, notice how combining the terms [latex]5x[/latex] and [latex]-4x[/latex] on the left leaves us with an equation with exactly the same terms on both sides of the equal sign.

You arrive at the true statement “[latex]3=3[/latex].” When you end up with a true statement like this, it means that the solution to the equation is “all real numbers.” Try substituting [latex]x=0[/latex] into the original equation—you will get a true statement! Try [latex]x=-\frac{3}{4}[/latex], and it also will check!

This equation happens to have an infinite number of solutions. Any value for x that you can think of will make this equation true. When you think about the context of the problem, this makes sense—the equation [latex]x+3=3+x[/latex] means “some number plus 3 is equal to 3 plus that same number.” We know that this is always true—it’s the commutative property of addition!

In the following video, we show more examples of attempting to solve a linear equation with either no solution or many solutions.

In the following video, we show more examples of solving linear equations with parentheses that have either no solution or many solutions.

Absolute value equations with no solutions

As we are solving absolute value equations it is important to be aware of special cases. An absolute value is defined as the distance from 0 on a number line, so it must be a positive number. When an absolute value expression is equal to a negative number, we say the equation has no solution, or DNE. Notice how this happens in the next two examples.

In this last video, we show more examples of absolute value equations that have no solutions.

Summary

Complex, multi-step equations often require multi-step solutions. Before you can begin to isolate a variable, you may need to simplify the equation first. This may mean using the distributive property to remove parentheses or multiplying both sides of an equation by a common denominator to get rid of fractions. Sometimes it requires both techniques. If your multi-step equation has an absolute value, you will need to solve two equations, sometimes isolating the absolute value expression first.

We have seen that solutions to equations can fall into three categories:

• One solution
• No solution, DNE (does not exist)
• Many solutions, also called infinitely many solutions or All Real Numbers

And sometimes, we don’t need to do much algebra to see what the outcome will be.